I want to make sure in no way am I liable for any issues, and that my code cannot simply be copied and claimed as someone else's.
You want a license that includes a disclaimer of liability (often in ALL CAPS) and requires maintaining attribution. Virtually any mainstream open source license will include both of these. The MIT license is perfectly suitable and doesn't contain any other baggage, either. You might also be interested in the Apache 2.0 license, which is quite similar but licenses patent rights as well as copyright rights. (This is helpful for recipients, since they know you won't sue them for patent infringement later.)
Do I have to apply or register with some kind of group or organisation to be able to use a license given in the list of GitHub templates, or can I just slap a template on my work?
Any open source license you see is free to use; you will not encounter any legal trouble if you use any mainstream OSI-approved license. The texts of these licenses are, I guess, technically copyrighted, but their authors deliberately offer them to the public exactly so that you can include them in your projects. Please do that!
Another thing I notice is that the MIT license template sticks a Copyright notice on the work. Do I have to apply for this copyright in any way, or is slapping it on once again enough for it to be valid?
In the 172 nations that are signatories of the Berne Convention, copyright automatically applies to a work at the moment it is fixed in a tangible form. Your work does have copyright rights attached because it is an original, tangibly expressed creative work. You are entitled to say so via a copyright notice.
Note that in some jurisdictions there are benefits to officially registering your copyright. In the United States if someone infringes your copyright on a work, you can only seek actual damages if the work was not registered with the U.S. Copyright Office at the time the infringement took place (i.e., you lost X dollars of potential sales and are demanding it back). If the work was registered during the infringement, you can seek statutory damages (i.e., punitive money that may be higher than the amount of actual money involved in the infringement).
Lastly, is the Full Name given under the Copyright notice valid if I use my internet name, DevelopedLogic, or do I have to apply my actual, real-life name?
U.S. law does not require real names on copyright registrations, so I assume it wouldn't require them on copyright notices either. You may make things more difficult on yourself if you need to legally enforce your license, though, since you will need to take the extra step of associating your real identity with the pseudonym for the court proceedings. If, on balance, you think the small likelihood of that legal headache actually happening is less worrying than revealing your real name on the Internet, then go ahead and use your pseudonym.
